I think he can get iron from spinach and possibly other vegetables. What I would be worried about also is vitamin B12. This can be gotten from eggs and milk, but may be hard to digest. I think a supplement is best.
A diet low in red meat has advantages. Also, a lot of fish has advantages too, especially wild Salmon, however you need to be worried about mercury intake.
I know it sounds complicated, but once you learn you can give him a good diet. I think overall the tendency to stay away from red meat is a healthy thing--but entirely away from it is challenging.
Wikipedia is a good source for looking up iron and vitamin B12 sources other than red meat. You could also find out which fishes are less dangerous in terms of mercury. If you figure it out, this diet could have a healthy effect on your entire family.
From what I have seen (I'm not a parent, but have about 30 nieces and nephews), pressuring a child to eat something will just make him resist more. Acting like it's nothing, and working around it keeps his "honor" and he may go back to it on his own, but overall this is much better than, say, refusing to eat anything but deserts. Then you'd have a challenge. Less red meat is a good direction.
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